EXCHANGE 


AUG  17  i 


Observation  and  Reduction  of  Occultations  of 
Stars  by  the  Moon 

WITH  A  DETERMINATION  OF  THE  RESULTING  LONGITUDE  OF 

THE  FLOWER  OBSERVATORY,  AND  CORRECTIONS  TO  THE 

RIGHT  ASCENSION,  DECLINATION  AND  SEMI-DIAMETER 

OF  THE  MOON 


BY 


KRIKORIS  GARABED  BOHJELIAN 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 

UNIVERSITY  OF  PENNSYLVANIA  IN  PARTIAL  FULFILMENT 

OF  THE  REQUIREMENTS  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 


UN!\ 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


1915 


Observation  and  Reduction  of  Occupations  of 
Stars  by  the  Moon 

WITH  A  DETERMINATION  OF  THE  RESULTING  LONGITUDE  OF 

THE  FLOWER  OBSERVATORY,  AND  CORRECTIONS  TO  THE 

RIGHT  ASCENSION,  DECLINATION  AND  SEMI-DIAMETER 

OF  THE  MOON 


BY 

KRIKORIS  GARABED  BOHJELIAN 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 

UNIVERSITY  OF  PENNSYLVANIA  IN  PARTIAL  FULFILMENT 

OF  THE  REQUIREMENTS  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


1915 


INTRODUCTION. 


It  is  well  known  that  corrections  to  the  coordinates,  distance, 
and  size  of  the  Moon  can  be  determined  from  the  observations 
of  occultations  more  accurately  than  from  any  other  method. 

If  simultaneous  observations  of  this  kind  are  secured  from  two 
stations  on  the  Earth,  which  differ  widely  in  latitude,  the  oblate- 
ness  of  the  Earth  can  also  be  found ;  and  whatever  the  situation 
of  the  stations,  their  difference  in  longitude  can  be  thus  deter- 
mined with  a  higher  accuracy  than  by  any  other  method,  except 
that  of  the  Telegraph  and  Wireless. 

As  the  Longitude  of  the  Flower  Observatory  was  accurately 
determined,  both  by  Telegraph  and  Wireless  method,  a  com- 
parison of  these  results  with  a  value  found  from  occultations 
becomes  of  interest,  and  as  the  later  observations  have  a  special 
value  for  improving  our  knowledge  of  the  Moon's  motion,  the 
following  piece  of  work  was  undertaken  with  these  objects. 

The  work  was  begun  in  the  early  summer  of  1914,  the  obser- 
vations being  made  with  the  1 8-inch  equatorial  of  the  Flower 
Observatory  of  the  University  of  Pennsylvania. 

Among  the  occultations  observed  it. was  learned  that  13  had 
been  simultaneously  observed  by  Prof.  Asaph  Hall,  with  the 
26-inch  equatorial  of  the  U.  S.  Naval  Observatory. 

Through  the  courtesy  of  the  Director,  Captain  J.  A.  Hooge- 
werff,  these  observations  were  forwarded  to  Prof.  Eric  Doolittle. 

It  is  the  results  from  these  stars,  which  form  the  principal 
basis  of  the  following  investigation. 

OBSERVED  TIMES  OF  OCCULTATIONS. 


Date,  1914 

Star 

Phase 

Phila.  M.  T. 

Washington  M.  T. 

July   17 

q  Tauri 

I 

I3h  59m  33s-4 

I3h  5im    6s.o 

20 

I 

14     n     18  .0 

14      2     33  .6 

16 

E 

H    38    57-8 

14    29      4.1 

q 

E 

14    57      4.2 

H    47     37-5 

20 

E 

15       6     57.8 

14    57    47-1 

21 

E 

15       19         3  -2 

22 

E 

15     22     23  .2 

A 

g- 

r 

A  Aquarii 

I 

-   ii       7     52  .9 

10    57     23  .6 

11 

E 

12     28     19  .9 

12     16    25  .6 

78 

I 

12      46      42  .4 

12     34    35-3 

78         " 

E 

14      7      2  .7 

13     55     52  .4 

30 

r  Sagit 

I 

9     14     18  .2 

9       3     50  .6 

«      « 

E 

10    30     18  .9  ' 

10    20     53  .3 

Sept.  14 

K  Gemin. 

I 

14     16       3  .2 

14      8     20  .9 

E 

14     50      o  .4 

H    39     3°  -o 

311753 


OBSERVATION  AND  REDUCTION  OF 


PRELIMINARY  COMPUTATION. 

The  right  ascension  and  declination  of  the  above  stars,  reduced 
to  apparent  place  for  the  observed  times  are  as  follows: 


Date,  1914 

Name 

Right  Ascension,  a 

Declination,  8 

July   17 

« 

« 
Aug.    7 

30 
Sept.  14 

q  Tauri 

20       " 

16      " 

21        " 

22        " 

X  Aquarii 

78  c  •; 

T  Sagittarn 
ic  Geminorum 

55°     i'   39"-45 
55     10    58  .50 
54    55    42  .11 
55     12       5  -68 
55     14     12  .27 
342      2     49  .65 
342    32     15  .38 
285    24    35  .99 
114    49     28  .89 

+24°  12'     4".72 
24      6      9.  90 

24         I      21    .47 
24      17      23    .16 

+24     15     47  .93 
—  8       i     56  .12 
-  7    39    24  .26 
-27     47     53  -78 
+24    36     19  .74 

The  right  ascension,  declination  and  horizontal  parallax  of  the 
Moon  for  consecutive  hours  on  the  successive  dates  are  found 
to  have  the  following  values: 


Date,  1914 

A 

D 

7T 

July  17 

i8h 

53° 

29' 

14 

".70 

+24° 

27' 

24 

".0 

54' 

48".98 

19 

54 

i 

31 

•05 

34 

59 

.8 

50  .03 

20 

54 

33 

52 

•45 

42 

29 

.0 

51  .10 

21 

55 

6 

19 

•50 

49 

.6 

52  .17 

22 

55 

38 

•75 

+24 

57 

7 

•4 

54 

53  .26 

Aug.  7 

15 

340 

49 

21 

.90 

-  7 

45 

44 

•9 

55 

28  .89 

16 

341 

17 

20 

.40 

7 

31 

24 

•3 

27  .56 

17 

341 

45 

15 

.60 

7 

17 

2 

•7 

26  .28 

18 

342 

13 

7 

•65 

7 

2 

40 

.1 

25  .01 

19 

342 

40 

56 

.70 

-  6 

48 

16 

•5 

55 

23  -74 

Aug.  30 

13 

284 

30 

16 

.95 

-27 

I 

29 

•5 

57 

25  .78 

H 

285 

7 

12 

.30 

26 

56 

.7 

24  -47 

15 

285 

44 

2 

•55 

26 

50 

50 

.6 

23  .15 

16 

286 

20 

47 

.70 

-26 

45 

17 

.2 

57 

21  .83 

Sept.  14 

18 

H3 

7 

15 

.90 

+25 

3i 

35 

.6 

56 

58  -49 

19 

113 

42 

37 

.05 

24 

18 

4 

57 

o  .71 

20   |  114 

17 

56 

.85 

16 

52 

.1 

57 

2  -94 

21     114 

53 

15 

.00 

+25 

9 

16 

.8 

57 

5  .18 

The  coordinates  of  the  Moon's  center  and  their  derivatives 
for  the  above  hours  are  computed  from  the  formulae: 


x 


cos  D  sin  (A  —  a) 


sin 


sin  (D  -  5)  cos2  \(A  -  a)  +  sin  (D  -  d)  sin2  \(A  -  a) 


sin  TT 


OCCULTATIONS  OF  STARS  BY  THE  MOON 


Date 

X 

x' 

y 

y' 

July  17 

q  Tauri 

i8h 

-I.53447I4 

+.5370561 

+0.2879661 

+•1335941 

19 

-  .9973448 

1910 

.4215411 

572 

20 

—  .4601054 

2817 

.5550831 

282 

21 

+  .0772044 

3309 

.6886000 

.1335060 

20  Tauri 

18 

—  1.6891416 

2214 

.3976049 

.1329862 

19 

-I.I5I8438 

3673 

.5305682 

409 

20 

—  O.6I442IO 

4714 

.6634894 

031 

21 

+0.0769149 

.5375338 

.7663772 

.1328740 

1  6  Tauri 

18 

-1.4356003 

.5369492 

.4824448 

.1339496 

19 

—  0.8985849 

.5370746 

.6163670 

8963 

20 

—  0.3614650 

1081 

.7502065 

8517 

21 

+O.I757I73 

2065 

.8840731 

.1338142 

21  Tauri 

19 

—  I.I70408l 

4028 

.3261183 

.1328847 

20 

-0.6329631 

4872 

.4590122 

8943 

21 

-0.0954338 

57H 

•5918895 

8515 

22 

+  0.4421798 

6558 

.7246927 

7562 

22  Tauri 

19 

-1.2053922 

4409 

•3553572 

7474 

2O 

—  0.6679063 

5296 

.4881108 

7515 

21 

-0.1303335 

6169 

.6208439 

7064 

22 

+  0.4073266 

.5377030 

.7535066 

.1326120 

Aug.  7 

X  Aquar. 

15 

—  I.3II9302 

.4987673 

.2898113 

.2599905 

16 

-0.8I3I459 

7982 

.5497641 

9144 

17 

-0.3I4340I 

8103 

.8096338 

8292 

18 

—  0.1844686 

8039 

1.0694310 

.2597498 

78  Aquar. 

16 

-I-339I549 

2319 

0.1422995 

.2602937 

17 

—  0.8408942 

2868 

.4025699 

2468 

18 

-0.3425869 

3251 

.6627924 

1979 

19 

+0.1557504 

4983468 

.9229651 

.2601472 

Aug.  30 

T  Sagit. 

13 

—  0.8425482 

.5723855 

.8049339 

.0945414 

14 

—  0.2701396 

4281 

.8994167 

4283 

15 

+0.3023008 

4491 

.9938027 

35i8 

16 

+0.8747514 

.5724485 

1.0881326 

+.0943142 

Sept.  14 

K  Gemin. 

18 

—  I.6l88750 

.5594691 

.9800360 

-.1344901 

19 

-1.0593263 

5259 

.8458102 

2051 

20 

-0.4998271 

5686 

.7115758 

2730 

21 

+0.0597563 

+-559595I 

+0.5772448 

-.1343938 

The  coordinates  and  their  derivatives  for  Philadelphia  and 
Washington  are  computed  from  the  formulae: 


p  sin 


b  sin  B 


p  cos  <pf  cos  (M  —  a)  =  b  cos  B 


=  p  cos  (pf  sin  (n  —  a) 
=  b  sin  (B  -  5) 


6  OBSERVATION  AND  REDUCTION  OF 

where  for  Phila.  <pr  =  39°  46'  32". 9  and  log  p  =  9.999400 
long.  =  5h  im  6s.5i 

and  for  Wash.   <?'  =  38°  55'  I4".o  and  log  p  =  9.999431  and 
long.  =  5h  8m  I5S78. 


7 

E 

1 

17 

f 

i? 

q  Tauri 

Phila. 
Wash. 

-.7674918 
-.7785898 

+.5816687 
+.5811267 

-.7427171 
.7634340 

+  .5034517 
.5062870 

20        " 

Phila. 
Wash. 

.7664782 

;  .7790062 

.5669897 
.5666500 

.7342088 
.7540638 

.4918996 
.4904268 

16      " 

Phila. 
Wash. 

.7556986 
.7722021 

.5289977 
•5287325 

21        " 

Phila. 

.7215053 

.4747025 

22        " 

Phila. 

.7180547 

47IIOI8 

X  Aqua. 

Phila. 
Wash. 

.4707500 
.5200407 

.7173353 
.6997824 

.83796466 
.2926039 

•7345912 
.7196153 

78      " 

Phila. 
Wash. 

.1979788 
—.2406685 

.7320044 
.7292446 

—  .0692538 
—  .0323282 

.7350480 
.7230056 

r  Sagit. 

Phila. 
Wash. 

+.1569340 
+.1214520 

.9155368 
.9116275 

+  .3904763 
+  .3611283 

.8737904 
.8746794 

K  Gemin. 

Phila. 
Wash. 

-.7668061 
-.7790125 

.5673483 
+.5643084 

-.7535621 
-.7705400 

.5202768 
+  .5202708 

With  the  assumed  value  of  the  longitude  of  Flower  Observatory 
viz:  5h  8m  6s.5i  and  of  the  U.  S.  Naval  Observatory  viz:  5h  8m 
15s.  78,  we  reduce  Phila.  and  Wash,  times  to  Greenwich  times 
and  assuming  the  values  of  t1  sufficiently  near  to  these  times,  so 
that  x  and  y  may  be  assumed  to  vary  uniformly  during  the 
interval  we  compute  certain  auxiliaries  first  introduced  by  Bessel 
by  the  formulae: 


m  sin  M  = 
m  cos  M  = 


n  sin  N  =  x' 
n  cos  N  =  y' 


sn 


sin  (M  -  N)  log  k  9435376o. 


After  obtaining  the  above  auxiliaries  viz:  log  m,  log  n,  M,  N, 


OCCULTATIONS  OF  STARS  BY  THE  MOON 

and  ^  we  next  compute  12,  T,  x,  v,  from  the  formulae: 

12  =  h  [-cos  4,  -  -cos  (M  -  N)  1  -  (/'  -  t)     h  =  3600 
In  n  J 


f-p  __    J.  J^ 

n 


sin  N  —  yo  cos 
—  #o  cos  N  —  yo  sin  N 


I 

1 

Phila. 

Wash. 

T 

log  x 

V 

q  Tauri 

I 

5h    jm    I9s79 

5h  8m  4I8.89 

+20.565 

9.81273 

+  1.98 

E 

1947 

35-82 

20.565 

9.81269 

i 

20        " 

I 

19  .OO 

33.84 

2O.79O 

9.89853 

< 

E 

19  .82 

42.42 

2O.79O 

9.89849 

< 

16      " 

E 

19-53 

40  .11 

20.315 

9.9H35 

' 

21        " 

E 

19.19 

2O.9II 

977634 

' 

22        " 

E 

20.28 

20.960 

9.80208 

1 

\  Aquar. 

I 

19.13 

32.51 

I6.83I 

9.93616 

;92 

E 

17  .68 

36.43 

17.762 

0.05258 

78      " 

I 

18.99 

15-63 

17-995 

9.87278 

<     . 

E 

18  .72 

41  .60 

17-995 

9.87080 

1 

T  Sagit. 

I 

1  8  .90 

37.78 

14.207 

9.96912 

1.  80 

E 

12  .67 

31.96 

I4.2O8 

9.96912 

« 

K  Gemin. 

I 

23-94 

32.74 

20.446 

O.OOII2 

1.82 

E 

5     i     23  .85 

5    8     32.49 

+20.506 

9.93839 

+  1.82 

The  coefficients  for  the  final  equations  are  obtained  from  the 
expressions : 

v  tan  \f/,    v.E  =  v(n(t  +  W  —  T)  —  tan  \l/x),    v  sec  \f/ 


Philadelphia 

Washington 

v  tan  \f/ 

v-E 

v  sec  »^ 

v  tan  ^ 

v-E 

v  sec  \ff 

q  Tauri 

I 

+0.779 

—  2.2OO 

-2.13 

+0.779 

—  2.24O 

—  2.130 

E 

+O.I3I 

-0.732 

+  1.98 

—  0.188 

-0.815 

+  1.980 

20        " 

I 

+0.423 

—  2.140 

—  2.  02 

+0.385 

—  2.070 

—  2.O2 

E 

—  1.160 

+O.2O7 

+  2.29 

—  1.  120 

+0.134 

+2.27 

16      " 

E 

-0.963 

+0.073 

+2.  2O 

—  O.926 

+O.OO4 

+2.18 

21         " 

E 

+0.267 

—  0.782 

+  1-99 

22        " 

E 

-0.033 

-0.595 

+  1.98 

X  Aquari 

I 

+0.068 

-0.751 

-1.92 

—  O.OI7 

—  0.8II 

-1.92 

E 

-0.732 

—  O.O92 

+4.06 

-0.673 

+  I.2IO 

+2.04 

78      " 

I 

+0.040 

—  O.24O 

-i-93 

+  0.083 

-P-365 

-1.94 

E 

—  I.OIO 

+  1.990 

+2.18 

-0.946 

+  1.870 

+2.05 

r  Sagit. 

I 

+0.361 

—  O.27I 

—  1.84 

+0.342 

-0-335 

-1.83 

E 

—  I.OIO 

+2.320 

+2.06 

-0.958 

+2.230 

+2.04 

K  Gemin. 

I 

+1.960 

-3.180 

-2.67 

+2.090 

-3.330 

-2.77 

E 

-3.880 

+2.970 

+4.28 

-4470 

+3-130 

+4.82 

8  OBSERVATION  AND  REDUCTION  OF 

FORMATION   OF  THE   FINAL  EQUATIONS  OF 
CONDITIONS. 

Writing  the  results  thus  far  obtained,  we  may  now  set  up  the 
following  equations,  which  we  divide  into  four  groups : 


W  =  5h 
W  = 
W  = 
W  = 

W  = 
W  = 

W  =  5 
W  = 
W  = 
W  = 
W  =  5 


July  17. 

Group  I 

/I 

im 

19* 

'.79  - 

1.987 

+  0.7791? 

—  2.I37TAK 

—  2.2OOA?r 

(2) 

19 

•49  ~ 

1.98 

+  0.131 

+   I-98 

-  0.732 

(4) 

19 

.00  — 

1.98 

+  0.423 

—  2.  02 

—  2.140 

(6) 

19 

.82  - 

1.98 

—  1.160 

+  2.29 

+  0.207 

(8) 

19 

•53  - 

1.98 

-  0.963 

+  2.  2O 

+  0.073 

(10) 

19 

.19  - 

1.98 

+  0.267 

+  1-99 

—  0.782 

(30) 

i 

20  .28  — 

1.98 

-  0.033 

+  1.98 

-  0.595 

(40) 

8 

41 

.89- 

1.98 

+  0.799 

-2.I3 

—  2.240 

(i) 

35 

•83- 

1.98 

+  0.188 

+  1.99 

-  0.815 

(3) 

33 

.84- 

1.98 

+  0.385 

—  2.OI 

—  2.070 

(5) 

42 

.42  - 

1.98 

—  1.  120 

+  0.27 

+  0.134 

(7) 

8 

40 

.11  — 

1.98 

—  0.926 

+  2.18 

+  0.004 

(9) 

Aug.  7. 

Group  II'  i 

W 

=  5b 

im  I98.I3 

—  i 

.927  +  o.o68t?  —  i.937rA/c  —  O.75IA7T 

(12) 

W 

= 

17 

.69 

—  i 

.92 

+  0.732 

+  2.06 

—  0.092 

(14) 

W 

= 

18 

.99 

—  i 

.92 

+  0.040 

—  i 

•93 

—  0.240 

(16) 

w 

=  5 

i     18 

.72 

—  i 

.92 

—  I.OIO 

+  2 

.18 

+  1.990 

(18) 

w 

=  5 

8    32 

•51 

-  1.92 

+  0.017 

—  I 

.92 

—  0.811 

(ii) 

w 

= 

36 

•43 

—  i 

.92 

-  0.673 

+  2.O4 

+  I.2IO 

(13) 

w 

= 

15 

•63 

—  i 

.92 

+  0.084 

—  I 

•94 

~  0.365 

(15) 

w 

=  5 

8     41 

.60 

—  i 

.92 

—  0.946 

—  2 

•05 

+  1.870 

(17) 

Aug.  30.     Group  III'  i 

W  =  5h   im  i88.90  —  1.807  +  0.361$  —  i.847rA/c  —  o.27iA?r  (20) 

W  =  5     i     12  .67  —  1.80    —  i.oio    +  4.06  +  2.320  (22) 

W  =  5     8    37  .78  -  i  .80    +  0.342     -  1.83  -  0.335  (19) 

W'  =  5     8     31  .96  —  i.  80    —  0.958    +  2.04  +  2.230  (21) 

Sept.  14.     Group  IV7  i 

W  =  5h   im  238.94  —  1.827  +  1.96??  —  2.67?rAK  —  3.i8A7r  (24) 

W  =  5     i     23  .85  -  1.82    -  3.88      +  4.28  +  2.97  (26) 

W  =  5     8    32  .74  -  1.82    +  2.09      -  2.77  -  3.33  (23) 

W  =  5    8     32  .49  -  1.82     -  4.47      +  4-82  +  3.13  (25) 


If  we  assume  7,  &,  ATT,  and  irAk  to  be  the  same  in  all  of  these 
four  groups  —  an  assumption  which  involves  no  appreciable  error 
—  we  shall  have  26  equations  in  four  groups,  between  those 
quantities  and  w  and  w'.  The  longitude  of  Washington  (5h  8m 
1  5s.  78)  however,  will  be  considered  to  be  correctly  obtained. 


OCCULTATIONS  OF  STARS  BY  THE  MOON  9 

It  is  evident,  however,  that  for  various  reasons  a  direct 
solution  of  these  equations  in  each  group  will  not  be  expedient. 
In  the  first  place,  the  large  terms  involved  would  render  the 
operation  very  laborious,  and  furthermore,  it  will  not  be  possible 
to  separate  ATT  from  the  remaining  quantities,  without  assuming 
both  w  and  wr  to  be  known. 

We  therefore  proceed  as  follows:  Assuming  the  equations  of 
equal  weight,  we  subtract  the  first  from  the  third,  the  third  from 
the  fifth,  etc.;  and  the  fourth  from  the  second,  the  sixth  from  the 
fourth,  etc.;  continuing  thus  we  obtain  the  following  groups  of 
equations : 

Group  I'  2 


4- 

0.6481? 

— 

4.II07TA/C 

=  4- 

i. 

47OA7T 

-    0.32 

4- 

0.292 

— 

3.000 

=  4-  1.410 

4-   0.47 

4- 

1.580 

- 

4.310 

=  4-  2.350 

4-    0.82 

4- 

0.195 

— 

0.092 

=  4-  0.134 

4-   0.29 

— 

1.740 

4- 

4-320 

=  —  2.280 

4-   0.26 

4- 

0.611 

— 

4-I2O 

=  4-  1.420 

—    6.07 

4- 

0.197 

— 

4.OOO 

=  4-  1.850 

4-    1.98 

4- 

1.510 

— 

4.290 

=  4-  2.  200 

4-   8.58 

4- 

0.195 

— 

0.089 

=  + 

o. 

130 

4-   2.31 

+ 

1.730 

+ 

4.320 

=   —  2.210 

4-    1.78 

Group  II'  2 

4- 

o.Sootf 

4- 

3-997rA« 

=  4- 

0.659A7T 

-    144 

— 

0.772 

+  3-99 

=  4- 

0.148 

4-    1.30 

4- 

1.050 

— 

4.10 

=  4-  2.230 

-   0.27 

4- 

1.080 

4-4-io          =  - 

2.740 

4-   0.41 

4- 

0.690 

— 

3.96 

=   4-  2.O2O 

4-   3-92 

- 

0.756 

4- 

3-98 

-  -  1.580 

—  20.80 

4- 

1.030 

— 

3-99 

=  4-  2.240 

4-  25.90 

— 

0.963 

4- 

2.97 

=  —  2.680 

-   9-09 

Group  III'  2 

+  1.37$    —  3.907rA»c    =  +  2.59AX    —    6.23 
+  1.30       -  3-88  =  +  2.56         -    5.82 

Group  IV'  2 

+  5.841?    —  6.96?rAK    =  +  6.I5A7T    —    0.09 
+  6.56       -  7.60  =  +  6.47         -    0.25 

By  means  of  these  four  groups  of  equations  of  condition,  viz: 
(I'2,  II'2,  IH'2,  IV'2)  we  now  determine  the  most  probable 
values  of  &  and  ATT£.  The  value  of  ATT,  however,  cannot  be 
well  determined,  as  we  have  before  remarked,  If  it  were  not 


IO  OBSERVATION  AND   REDUCTION  OF 

known  a  priori  that  such  was  the  case,  it  would  be  shown  from 
the  normal  equations,  which  would  be  practically  indeterminate 
for  this  quantity. 

We  should,  therefore,  determine  &  and  nAk  in  terms  of  ATT, 
in  order  to  see  what  effect  an  error  in  TT  will  have  upon  the 
longitude. 

We  derive  from  the  above  equations,  for  groups  I '2  and  1 1  '2, 
only,  the  following  two  sets  of  normal  equations;  the  last  two 
groups  are  solved  as  they  stand,  since  there  are  two  equations 
in  each  group. 

Normal  to  Group  I'  2  (or  I'  3) 

+  II.Slz?  —     2I.237TA/C  =  +  10.07  ATT  +     I4-5O 

—  21.231?  +  i33.057rA/c  =  —  62.5IA7T  —    14.89 

Normal  to  Group  IT  2  (or  11'  3) 
+    6.54$  —    12.47^    =  +    7.27A?r  +    51.9 

—  12.40??  +  I28.6TA/C     =  —  48.3OA7T  —  186.6 

From  I '3  we  obtain 

&  =  +  1.428  +  .ooSATT 
?rA&  =  +  o.i  12  —  470A7T 

To  find  7  we  now  substitute  these  values  in  (i),  (3),  (5),  (7), 
and  (9),  and  observing  that  wf  —  5h  8m  i5s-78,  we  find  the  mean 
value  of  7  to  be 

7  =  +  6".52  -  .639A7T 

We  now  substitute  these  values  of  #,  7rA&,  7  in  (2),  (4),  (6), 
(8),  (10),  (30),  (40),  when  we  find  the  following  values  for  the 
difference  of  longitude  between  Greenwich  and  the  Flower 
Astronomical  Observatory  of  the  University  of  Pennsylvania: 

W    =    5*1    jm  78^7   _j_    .07A7T 

w  =  5  i  6  .99  —  .40 

w  =  5  i  6  .49  -  .07 

w  =  5  i  6.  53  +  .39 

w  =  5  i  6  .51  +  .29 

w  =  5  i  6  .89  —  .48 

™  =  5  i  7  -54  ~  -27 

Mean     w  =  5h  im  6S.96  —  .o6A7r 


OCCULTATIONS  OF  STARS  BY  THE  MOON 


II 


And  the  resulting  longitude  from  Washington  is 

\    =    —   7m  8S.82    -    .06A7T 

With  the  above  values  of  y  and  &  we  may  now  determine  correc- 
tions to  the  assumed  right  ascension  and  declination  of  the  Moon. 
We  have  the  formulae : 

sin  N  cos  D  •  Aa  +  cos  N.     A5  =  7 
—  cos  N  cos  D  •  Aa  —  sin  N.     A5  =  # 
and  from  these 

Aa  =  +  6".58        A5  =  +-  2".Q6 

Assuming  the  errors  of  the  star  places  to  be  inappreciable,  these 
will  represent  the  errors  in  the  computed  right  ascension  and 
declination  of  the  Moon  at  a  time  corresponding  to  the  mean  of 
times  of  the  observations. 

These  corrections,  it  will  be  seen,  are  affected  by  any  small 
outstanding  error  in  the  parallax,  as  they  have  been  derived  by 
assuming  ATT  =  o. 

In  the  same  way,  assuming  Ax  =  o  and  taking  the  mean  of  the 
values  given  above,  viz:  329 1"  we  find  from  the  above  value  of 

-f-  ."ii 2 

A&  =  -f-  .000034 


we  have  assumed 
therefore, 


k  =  +  .272506 
k  =  +  .272540 


as  shown  from  these  observations. 

In  the  same  way  by  solving  the  other  groups  of  equations,  we 
obtain  the  following  results: 


Group 

*Lk 

tf 

7 

Aa 

I 

II 
III 

IV 

+".H2—  O.470A7T 
—   .839—0.328 
—   .500—0.560 
—  .322—9.700 

+  l".428+00.008A7T 

+6  .350+00.500 
-5  .960+00.300 
—3  .820  —  10.500 

+  6".52-.639A7r 
+  7  .47  +.320 
+  u   .59+440 
+  9  -93  +.380 

+  6".58 
+  3  -72 

+13  .92 

+  11  .01 

Group 

A5 

k 

IV 

X 

I 
II 
III 

IV 

+2.96 
+9.08 

-3-99 

—  1.22 

.272540 
.272408 
.272386 
.272415 

5h  i 
5     i 
5     i 
5     i 

m  68.96—  o.o6A7T 
6  .92—0.49 
6.86—0.80 
6  .89  —  I.48A7T 

_7m 

-7 
-7 
-7 

83.82—  0.o6Air 

8  .86—0.49 
8  .92—0.80 
8  .89  —  I-48A7T 

Mean 

.272437 

5h  i 

m  68.9i—  O.43A7T 

_7m 

8S.87—  o.43A?r 

12     OBSERVATION  AND  REDUCTION  OF  OCCULTATIONS  OF  STARS 

CONCLUSION. 

The  errors  Aa  and  A5  in  the  Moon's  position  are  somewhat 
smaller  than  was  to  be  expected,  and  indicate  that  this  body  is 
following  its  computed  path  somewhat  more  closely  than  in 
recent  years. 

The  corrections  A&  to  the  apparent  semi-diameter  is  markedly 
negative,  but  it  is  possible  that  values  of  this  quantity  secured 
from  occultations  may  be  influenced  by  the  aperture  of  the 
instrument  employed. 

The  final  mean  value  of  longitude  of  Flower  Observatory  from 
U.  S.  Naval  Observatory,  as  shown  above,  from  this  work  is 

X  =  —  7*  88.87 

The  results  previously  obtained  for  the  same  quantity  are: 
By  Telegraph  X  =  -  ym  8s.9i 

By  Wireless  X  =  -  7m  8*74 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

RENEWALS  ONLY— TEL.  NO.  642-3405 
This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


SANTA  CRUZ 

INTERLI3RARY  LO 

\N                i 

FEB  5    1970 

!      MAR  9 

i  d 

p 

I 

1 

1 

LD2lA-60m-6,'69 
(J9096slO)476-A-32 


General  Library 

University  of  California 

Berkeley 


Gaylord  Bros.       , 

Makers 
Syracuse,  N.  Y. 

W.JAN.  21,  1908 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


